Inverse-Limit Formulas and Stable-Range Rigidity for Cyclotomic Sums

We study truncation-compatible families F=(Fm)m≥1 over Q[z] through an inverse-limit formalism, and we evaluate them at the punctured cyclotomic cosine points αk,n=cos(2πk/n) with the specialization z=n−1. For symmetric families of uniformly bounded total x-degree ≤d, we prove a stable-range rigidity theorem: for all n≥d+2, the cosine-point evaluation factors through the finitely many punctured cosine power sums P1(n),…,Pd(n). In the purely polynomial case, this implies eventual polynomiality in n. We then extend the framework to include fixed-product factors and package their cosine-point contribution in multiplicative invariants MQ(n). In the stable range, the bounded-degree symmetric part collapses as before; any remaining cyclotomic dependence occurs only through these explicit product terms. Finally, we show that coefficient extraction from such products produces further bounded-degree symmetric families, and we apply this to complete symmetric functions hr evaluated at cosine points.